Title:
Interval scheduling and colorful independent sets

Abstract: Numerous applications in scheduling, such as resource allocation or steel
manufacturing, can be modeled using the NP-hard Independent Set problem (given
an undirected graph and an integer k, find a set of at least k pairwise
non-adjacent vertices). Here, one encounters special graph classes like 2-union
graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise
unions of an interval graph and a cluster graph), on which Independent Set
remains NP-hard but admits constant-ratio approximations in polynomial time. We
study the parameterized complexity of Independent Set on 2-union graphs and on
subclasses like strip graphs. Our investigations significantly benefit from a
new structural "compactness" parameter of interval graphs and novel problem
formulations using vertex-colored interval graphs. Our main contributions are:
1. We show a complexity dichotomy: restricted to graph classes closed under
induced subgraphs and disjoint unions, Independent Set is polynomial-time
solvable if both input interval graphs are cluster graphs, and is NP-hard
otherwise.
2. We chart the possibilities and limits of effective polynomial-time
preprocessing (also known as kernelization).
3. We extend Halld\'orsson and Karlsson (2006)'s fixed-parameter algorithm
for Independent Set on strip graphs parameterized by the structural parameter
"maximum number of live jobs" to show that the problem (also known as Job
Interval Selection) is fixed-parameter tractable with respect to the parameter
k and generalize their algorithm from strip graphs to 2-union graphs.
Preliminary experiments with random data indicate that Job Interval Selection
with up to fifteen jobs and 5*10^5 intervals can be solved optimally in less
than five minutes.

Comments:

This revision does not contain Theorem 7 of the first revision, whose proof contained an error